Riemann Hypothesis and Random Walks: the Zeta Case

Total Page:16

File Type:pdf, Size:1020Kb

Riemann Hypothesis and Random Walks: the Zeta Case Riemann Hypothesis and Random Walks: the Zeta case Andr´e LeClaira Cornell University, Physics Department, Ithaca, NY 14850 Abstract In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for 1 its L-function is valid to the right of the critical line <(s) > 2 , and the Riemann Hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a 1-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. arXiv:1601.00914v3 [math.NT] 23 May 2017 a [email protected] 1 I. INTRODUCTION There are many generalizations of Riemann's zeta function to other Dirichlet series, which are also believed to satisfy a Riemann Hypothesis. A common opinion, based largely on counterexamples, is that the L-functions for which the Riemann Hypothesis is true enjoy both an Euler product formula and a functional equation. However a direct connection between these properties and the Riemann Hypothesis has not been formulated in a precise manner. In [1, 2] a concrete proposal making such a connection was presented for Dirichlet L-functions, and those based on cusp forms, due to the validity of the Euler product formula to the right of the critical line. In contrast to the non-principal case, in this approach the case of principal Dirichlet L-functions, of which Riemann zeta is the simplest, turned out to be more delicate, and consequently it was more difficult to state precise results. In the present work we address further this special case. Let χ(n) be a Dirichlet character modulo k and L(s; χ) its L-function with s = σ + it. It satisfies the Euler product formula 1 1 −1 X χ(n) Y χ(pn) L(s; χ) = = 1 − (1) ns ps n=1 n=1 n where pn is the n-th prime. The above formula is valid for <(s) > 1 since both sides converge absolutely. The important distinction between principal verses non-principal characters is the following. For non-principal characters the L-function has no pole at s = 1, thus there exists the possibility that the Euler product is valid partway inside the strip, i.e. has 1 abscissa of convergence σc < 1. It was proposed in [1, 2] that σc = 2 for this case. In contrast, now consider L-functions based on principal characters. The latter character is defined as χ(n) = 1 if n is coprime to k and zero otherwise. The Riemann zeta function is the trivial principal character of modulus k = 1 with all χ(n) = 1. L-functions based on principal characters do have a pole at s = 1, and therefore have abscissa of convergence σc = 1, which implies the Euler product in the form given above cannot be valid inside the critical strip 0 < σ < 1. Nevertheless, in this paper we will show how a truncated version of 1 the Euler product formula is valid for σ > 2 . The primary aim of the work [1, 2] was to determine what specific properties of the prime numbers would imply that the Riemann Hypothesis is true. This is the opposite of the more well-studied question of what the validity of the Riemann Hypothesis implies 2 for the fluctuations in the distribution of primes. The answer proposed was simply based on the multiplicative independence of the primes, which to a large extent underlies their pseudo-random behavior. To be more specific, let χ(n) = eiθn for χ(n) 6= 0. In [1, 2] it was proven that if the series N X BN (t; χ) = cos (t log pn + θpn ) (2) n=1 p is O( N), then the Euler product converges for σ > 1 and the formula (1) is valid to the 2 p right of the critical line. In fact, we only need BN = O( N) up to logs (see Remark 1); p when we write write O( N), it is implicit that this can be relaxed with logarithmic factors. For non-principal characters the allowed angles θn are equally spaced on the unit circle, and it was conjectured in [2] that the above series with t = 0 behaves like a random walk due to p the multiplicative independence of the primes, and this is the origin of the O( N) growth. Furthermore, this result extends to all t since domains of convergence of Dirichlet series are always half-planes. Taking the logarithm of (1), one sees that log L is never infinite to the right of the critical line and thus has no zeros there. This, combined with the functional equation that relates L(s) to L(1−s), implies there are also no zeros to the left of the critical line, so that all zeros are on the line. The same reasoning applies to cusp forms if one also uses a non-trivial result of Deligne [2]. In this article we reconsider the principal Dirichlet case, specializing to Riemann zeta itself since identical arguments apply to all other principal cases with k > 1. Here all angles θn = 0, so one needs to consider the series N X BN (t) = cos(t log pn) (3) n=1 which now strongly depends on t. On the one hand, whereas the case of principal Dirichlet L- functions is complicated by the existence of the pole, and, as we will see, one consequently needs to truncate the Euler product to make sense of it, on the other hand BN can be estimated using the prime number theorem since it does not involve sums over non-trivial characters χ, and this aids the analysis. This is in contrast to the non-principal case, where, however well-motivated, we had to conjecture the random walk behavior alluded to above, so in this respect the principal case is potentially simpler. To this end, a theorem of Kac p (Theorem 1 below) nearly does the job: BN (t) = O( N) in the limit t ! 1, which is also a consequence of the multiplicative independence of the primes. This suggests that one can 3 also make sense of the Euler product formula in the limit t ! 1. However this is not enough for our main purpose, which is to have a similar result for finite t which we will develop. This article is mainly based on our previous work [1, 2] but provides a more detailed analysis and extends it in several ways. It was suggested in [1] that one should truncate the series at an N that depends on t. First, in the next section we explain how a simple group structure underlies a finite Euler product which relates it to a generalized Dirichlet series which is a subseries of the Riemann zeta function. Subsequently we estimate the error under truncation, which shows explicitly how this error is related to the pole at s = 1, as expected. The remainder of the paper, sections IV-VI, presents various applications of these ideas. We use them to study the argument of the zeta function. We present an algorithm to calculate very high zeros, far beyond what is currently known. We also study the statistical fluctuations of individual zeros, in other words, a 1-point correlation function. In many respects, our work is related to the work of Gonek et. al. [4, 5], which also considers a truncated Euler product. The important difference is that the starting point in [4] is a hybrid version of the Euler product which involves both primes and zeros of zeta. Only after assuming the Riemann Hypothesis can one explain in that approach why the truncated product over primes is a good approximation to zeta. In contrast, here we do not assume anything about the zeros of zeta, since the goal is to actually understand their location. We are unable to provide fully rigorous proofs of some of the statements below, however we do provide supporting calculations and numerical work. In order to be clear on this, below \Proposal" signifies the most important claims that we could not rigorously prove. II. ALGEBRAIC STRUCTURE OF FINITE EULER PRODUCTS The aim of this section is to define properly the objects we will be dealing with. In par- ticular we will place finite Euler products on the same footing as other generalized Dirichlet series. The results are straightforward and are mainly definitions. Definition 1. Fix a positive integer N and let fp1; p2; : : : pN g denote the first N primes where p1 = 2. From this set one can generate an abelian group QN of rank N with elements n o n1 n2 nN QN = p1 p2 ··· pN ; ni 2 Z 8i (4) 4 + + where the group operation is ordinary multiplication. Clearly QN ⊂ Q where Q are the positive rational numbers. There are an infinite number of integers in QN which form a subset of the natural numbers N = f1; 2;:::g. We will denote this set as NN ⊂ N, and elements of this set simply as n. Definition 2. Fix a positive integer N. For every integer n 2 N we can define the character c(n): c(n) = 1 if n 2 NN ⊂ QN = 0 otherwise (5) Clearly, for a prime p, c(p) = 0 if p > pN .
Recommended publications
  • Euler Product Asymptotics on Dirichlet L-Functions 3
    EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS IKUYA KANEKO Abstract. We derive the asymptotic behaviour of partial Euler products for Dirichlet L-functions L(s,χ) in the critical strip upon assuming only the Generalised Riemann Hypothesis (GRH). Capturing the behaviour of the partial Euler products on the critical line is called the Deep Riemann Hypothesis (DRH) on the ground of its importance. Our result manifests the positional relation of the GRH and DRH. If χ is a non-principal character, we observe the √2 phenomenon, which asserts that the extra factor √2 emerges in the Euler product asymptotics on the central point s = 1/2. We also establish the connection between the behaviour of the partial Euler products and the size of the remainder term in the prime number theorem for arithmetic progressions. Contents 1. Introduction 1 2. Prelude to the asymptotics 5 3. Partial Euler products for Dirichlet L-functions 6 4. Logarithmical approach to Euler products 7 5. Observing the size of E(x; q,a) 8 6. Appendix 9 Acknowledgement 11 References 12 1. Introduction This paper was motivated by the beautiful and profound work of Ramanujan on asymptotics for the partial Euler product of the classical Riemann zeta function ζ(s). We handle the family of Dirichlet L-functions ∞ L(s,χ)= χ(n)n−s = (1 χ(p)p−s)−1 with s> 1, − ℜ 1 p X Y arXiv:1902.04203v1 [math.NT] 12 Feb 2019 and prove the asymptotic behaviour of partial Euler products (1.1) (1 χ(p)p−s)−1 − 6 pYx in the critical strip 0 < s < 1, assuming the Generalised Riemann Hypothesis (GRH) for this family.
    [Show full text]
  • Of the Riemann Hypothesis
    A Simple Counterexample to Havil's \Reformulation" of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 10025 [email protected] The Riemann Hypothesis (RH) is \the greatest mystery in mathematics" [3]. It is a conjecture about the Riemann zeta function. The zeta function allows us to pass from knowledge of the integers to knowledge of the primes. In his book Gamma: Exploring Euler's Constant [4, p. 207], Julian Havil claims that the following is \a tantalizingly simple reformulation of the Rie- mann Hypothesis." Havil's Conjecture. If 1 1 X (−1)n X (−1)n cos(b ln n) = 0 and sin(b ln n) = 0 na na n=1 n=1 for some pair of real numbers a and b, then a = 1=2. In this note, we first state the RH and explain its connection with Havil's Conjecture. Then we show that the pair of real numbers a = 1 and b = 2π=ln 2 is a counterexample to Havil's Conjecture, but not to the RH. Finally, we prove that Havil's Conjecture becomes a true reformulation of the RH if his conclusion \then a = 1=2" is weakened to \then a = 1=2 or a = 1." The Riemann Hypothesis In 1859 Riemann published a short paper On the number of primes less than a given quantity [6], his only one on number theory. Writing s for a complex variable, he assumes initially that its real 1 part <(s) is greater than 1, and he begins with Euler's product-sum formula 1 Y 1 X 1 = (<(s) > 1): 1 ns p 1 − n=1 ps Here the product is over all primes p.
    [Show full text]
  • The Mobius Function and Mobius Inversion
    Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Number Theory Mathematics via Primary Historical Sources (TRIUMPHS) Winter 2020 The Mobius Function and Mobius Inversion Carl Lienert Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_number Part of the Curriculum and Instruction Commons, Educational Methods Commons, Higher Education Commons, Number Theory Commons, and the Science and Mathematics Education Commons Click here to let us know how access to this document benefits ou.y Recommended Citation Lienert, Carl, "The Mobius Function and Mobius Inversion" (2020). Number Theory. 12. https://digitalcommons.ursinus.edu/triumphs_number/12 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Number Theory by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. The Möbius Function and Möbius Inversion Carl Lienert∗ January 16, 2021 August Ferdinand Möbius (1790–1868) is perhaps most well known for the one-sided Möbius strip and, in geometry and complex analysis, for the Möbius transformation. In number theory, Möbius’ name can be seen in the important technique of Möbius inversion, which utilizes the important Möbius function. In this PSP we’ll study the problem that led Möbius to consider and analyze the Möbius function. Then, we’ll see how other mathematicians, Dedekind, Laguerre, Mertens, and Bell, used the Möbius function to solve a different inversion problem.1 Finally, we’ll use Möbius inversion to solve a problem concerning Euler’s totient function.
    [Show full text]
  • Prime Factorization, Zeta Function, Euler Product
    Analytic Number Theory, undergrad, Courant Institute, Spring 2017 http://www.math.nyu.edu/faculty/goodman/teaching/NumberTheory2017/index.html Section 2, Euler products version 1.2 (latest revision February 8, 2017) 1 Introduction. This section serves two purposes. One is to cover the Euler product formula for the zeta function and prove the fact that X p−1 = 1 : (1) p The other is to develop skill and tricks that justify the calculations involved. The zeta function is the sum 1 X ζ(s) = n−s : (2) 1 We will show that the sum converges as long as s > 1. The Euler product formula is Y −1 ζ(s) = 1 − p−s : (3) p This formula expresses the fact that every positive integers has a unique rep- resentation as a product of primes. We will define the infinite product, prove that this one converges for s > 1, and prove that the infinite sum is equal to the infinite product if s > 1. The derivative of the zeta function is 1 X ζ0(s) = − log(n) n−s : (4) 1 This formula is derived by differentiating each term in (2), as you would do for a finite sum. We will prove that this calculation is valid for the zeta sum, for s > 1. We also can differentiate the product (3) using the Leibnitz rule as 1 though it were a finite product. In the following calculation, r is another prime: 8 9 X < d −1 X −1= ζ0(s) = 1 − p−s 1 − r−s ds p : r6=p ; 8 9 X <h −2i X −1= = − log(p)p−s 1 − p−s 1 − r−s p : r6=p ; ( ) X −1 = − log(p) p−s 1 − p−s ζ(s) p ( 1 !) X X ζ0(s) = − log(p) p−ks ζ(s) : (5) p 1 If we divide by ζ(s), the sum on the right is a sum over prime powers (numbers n that have a single p in their prime factorization).
    [Show full text]
  • 4 L-Functions
    Further Number Theory G13FNT cw ’11 4 L-functions In this chapter, we will use analytic tools to study prime numbers. We first start by giving a new proof that there are infinitely many primes and explain what one can say about exactly “how many primes there are”. Then we will use the same tools to proof Dirichlet’s theorem on primes in arithmetic progressions. 4.1 Riemann’s zeta-function and its behaviour at s = 1 Definition. The Riemann zeta function ζ(s) is defined by X 1 1 1 1 1 ζ(s) = = + + + + ··· . ns 1s 2s 3s 4s n>1 The series converges (absolutely) for all s > 1. Remark. The series also converges for complex s provided that Re(s) > 1. 2 4 P 1 Example. From the last chapter, ζ(2) = π /6, ζ(4) = π /90. But ζ(1) is not defined since n diverges. However we can still describe the behaviour of ζ(s) as s & 1 (which is my notation for s → 1+, i.e. s > 1 and s → 1). Proposition 4.1. lim (s − 1) · ζ(s) = 1. s&1 −s R n+1 1 −s Proof. Summing the inequality (n + 1) < n xs dx < n for n > 1 gives Z ∞ 1 1 ζ(s) − 1 < s dx = < ζ(s) 1 x s − 1 and hence 1 < (s − 1)ζ(s) < s; now let s & 1. Corollary 4.2. log ζ(s) lim = 1. s&1 1 log s−1 Proof. Write log ζ(s) log(s − 1)ζ(s) 1 = 1 + 1 log s−1 log s−1 and let s & 1.
    [Show full text]
  • First Year Progression Report
    First Year Progression Report Steven Charlton 12 June 2013 Contents 1 Introduction 1 2 Polylogarithms 2 2.1 Definitions . .2 2.2 Special Values, Functional Equations and Singled Valued Versions . .3 2.3 The Dedekind Zeta Function and Polylogarithms . .5 2.4 Iterated Integrals and Their Properties . .7 2.5 The Hopf algebra of (Motivic) Iterated Integrals . .8 2.6 The Polygon Algebra . .9 2.7 Algebraic Structures on Polygons . 11 2.7.1 Other Differentials on Polygons . 11 2.7.2 Operadic Structure of Polygons . 11 2.7.3 A VV-Differential on Polygons . 13 3 Multiple Zeta Values 16 3.1 Definitions . 16 3.2 Relations and Transcendence . 17 3.3 Algebraic Structure of MZVs and the Standard Relations . 19 3.4 Motivic Multiple Zeta Values . 21 3.5 Dimension of MZVs and the Hoffman Elements . 23 3.6 Results using Motivic MZVs . 24 3.7 Motivic DZVs and Eisenstein Series . 27 4 Conclusion 28 References 29 1 Introduction The polylogarithms Lis(z) are an important and frequently occurring class of functions, with applications throughout mathematics and physics. In this report I will give an overview of some of the theory surrounding them, and the questions and lines of research still open to investigation. I will introduce the polylogarithms as a generalisation of the logarithm function, by means of their Taylor series expansion, multiple polylogarithms will following by considering products. This will lead to the idea of functional equations for polylogarithms, capturing the symmetries of the function. I will give examples of such functional equations for the dilogarithm.
    [Show full text]
  • The Euler Product Euler
    Math 259: Introduction to Analytic Number Theory Elementary approaches II: the Euler product Euler [Euler 1737] achieved the first major advance beyond Euclid’s proof by combining his method of generating functions with another highlight of ancient Greek number theory, unique factorization into primes. Theorem [Euler product]. The identity ∞ X 1 Y 1 = . (1) ns 1 − p−s n=1 p prime holds for all s such that the left-hand side converges absolutely. Proof : Here and henceforth we adopt the convention: Q P The notation p or p means a product or sum over prime p. Q cp Every positive integer n may be written uniquely as p p , with each cp a nonnegative integer that vanishes for all but finitely many p. Thus the formal expansion of the infinite product ∞ Y X p−cps (2) p prime cp=0 contains each term Y −s Y n−s = pcp = p−cps p p exactly once. If the sum of the n−s converges absolutely, we may rearrange the sum arbitrarily and conclude that it equals the product (2). On the other hand, each factor in this product is a geometric series whose sum equals 1/(1 − p−s). This establishes the identity (2). The sum on the left-hand side of (1) is nowadays called the zeta function ∞ 1 1 X ζ(s) = 1 + + + ··· = n−s ; 2s 3s n=1 the formula (2) is called the Euler product for ζ(s). Euler did not actually impose the convergence condition: the rigorous treatment of limits and convergence was not yet available, and Euler either handled such issues intuitively or ignored them.
    [Show full text]
  • Elliptic Curves, Modular Forms, and L-Functions Allison F
    Claremont Colleges Scholarship @ Claremont HMC Senior Theses HMC Student Scholarship 2014 There and Back Again: Elliptic Curves, Modular Forms, and L-Functions Allison F. Arnold-Roksandich Harvey Mudd College Recommended Citation Arnold-Roksandich, Allison F., "There and Back Again: Elliptic Curves, Modular Forms, and L-Functions" (2014). HMC Senior Theses. 61. https://scholarship.claremont.edu/hmc_theses/61 This Open Access Senior Thesis is brought to you for free and open access by the HMC Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in HMC Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. There and Back Again: Elliptic Curves, Modular Forms, and L-Functions Allison Arnold-Roksandich Christopher Towse, Advisor Michael E. Orrison, Reader Department of Mathematics May, 2014 Copyright c 2014 Allison Arnold-Roksandich. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions. Contents Abstract iii Acknowledgments xi Introduction 1 1 Elliptic Curves 3 1.1 The Operation . .4 1.2 Counting Points . .5 1.3 The p-Defect . .8 2 Dirichlet Series 11 2.1 Euler Products .
    [Show full text]
  • Chapter 1 Euler's Product Formula
    Chapter 1 Euler’s Product Formula 1.1 The Product Formula The whole of analytic number theory rests on one marvellous formula due to Leonhard Euler (1707-1783): −1 X n−s = Y 1 − p−s . n∈N, n>0 primes p Informally, we can understand the formula as follows. By the Funda- mental Theorem of Arithmetic, each n ≥ 1 is uniquely expressible in the form n = 2e2 3e3 5e5 ··· , where e2, e3, e5,... are non-negative integers (all but a finite number being 0). Raising this to the power −s, n−s = 2−e2s3−e3s5−e5s ··· . Adding for n = 1, 2, 3,... , X n−s = 1 + 2−s + 2−2s + ··· 1 + 3−s + 3−2s + ··· 1 + 5−s + 5−2s + ··· ··· , each term on the left arising from just one product on the right. But for each prime p, −1 1 + p−s + p−2s + ··· = 1 − p−s , and the result follows. Euler’s Product Formula equates the Dirichlet series P n−s on the left with the infinite product on the right. 1–1 1.2. INFINITE PRODUCTS 1–2 To make the formula precise, we must develop the theory of infinite prod- ucts, which we do in the next Section. To understand the implications of the formula, we must develop the the- ory of Dirichlet series, which we do in the next Chapter. 1.2 Infinite products 1.2.1 Definition Infinite products are less familiar than infinite series, but are no more com- plicated. Both are examples of limits of sequences. Definition 1.1. The infinite product Y cn n∈N is said to converge to ` 6= 0 if the partial products Y Pn = cm → ` as n → ∞.
    [Show full text]
  • LECTURES on ANALYTIC NUMBER THEORY Contents 1. What Is Analytic Number Theory?
    LECTURES ON ANALYTIC NUMBER THEORY J. R. QUINE Contents 1. What is Analytic Number Theory? 2 1.1. Generating functions 2 1.2. Operations on series 3 1.3. Some interesting series 5 2. The Zeta Function 6 2.1. Some elementary number theory 6 2.2. The infinitude of primes 7 2.3. Infinite products 7 2.4. The zeta function and Euler product 7 2.5. Infinitude of primes of the form 4k + 1 8 3. Dirichlet characters and L functions 9 3.1. Dirichlet characters 9 3.2. Construction of Dirichlet characters 9 3.3. Euler product for L functions 12 3.4. Outline of proof of Dirichlet's theorem 12 4. Analytic tools 13 4.1. Summation by parts 13 4.2. Sums and integrals 14 4.3. Euler's constant 14 4.4. Stirling's formula 15 4.5. Hyperbolic sums 15 5. Analytic properties of L functions 16 5.1. Non trivial characters 16 5.2. The trivial character 17 5.3. Non vanishing of L function at s = 1 18 6. Prime counting functions 20 6.1. Generating functions and counting primes 20 6.2. Outline of proof of PNT 22 6.3. The M¨obiusfunction 24 7. Chebyshev's estimates 25 7.1. An easy upper estimate 25 7.2. Upper and lower estimates 26 8. Proof of the Prime Number Theorem 28 8.1. PNT and zeros of ζ 28 8.2. There are no zeros of ζ(s) on Re s = 1 29 8.3. Newman's Analytic Theorem 30 8.4.
    [Show full text]
  • Computational Number Theory in Relation with L-Functions 3
    Computational Number Theory in Relation with L-Functions Henri Cohen Abstract We give a number of theoretical and practical methods related to the com- putation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums), and in the global case (for instance Dirichlet L-functions, involving in particular the study of inverse Mellin transforms); we also give a number of little-known but very useful numerical methods, usually but not always related to the computation of L-functions. 1 L-Functions This course is divided into five parts. In the first part (Sections 1 and 2), we introduce the notion of L-function, give a number of results and conjectures concerning them, and explain some of the computational problems in this theory. In the second part (Sections 3 to 6), we give a number of computational methods for obtaining the Dirichlet series coefficients of the L-function, so is arithmetic in nature. In the third part (Section 7), we give a number of analytic tools necessary for working with L- functions. In the fourth part (Sections 8 and 9), we give a number of very useful numerical methods which are not sufficiently well-known, most of which being also related to the computation of L-functions. The fifth part (Sections 10 and 11) gives the Pari/GP commands corresponding to most of the algorithms and examples given in the course. A final Section 12 gives as an appendix some basic definitions arXiv:1809.10904v1 [math.NT] 28 Sep 2018 and results used in the course which may be less familiar to the reader.
    [Show full text]
  • Möbius Inversion from the Point of View of Arithmetical Semigroup Flows
    Biblioteca de la Revista Matematica´ Iberoamericana Proceedings of the \Segundas Jornadas de Teor´ıa de Numeros"´ (Madrid, 2007), 63{81 M¨obiusinversion from the point of view of arithmetical semigroup flows Manuel Benito, Luis M. Navas and Juan Luis Varona Abstract Most, if not all, of the formulas and techniques which in number theory fall under the rubric of “M¨obiusinversion" are instances of a single general formula involving the action or flow of an arithmetical semigroup on a suitable space and a convolution-like operator on functions. The aim in this exposition is to briefly present the general formula in its abstract context and then illustrate the above claim using an extensive series of examples which give a flavor for the subject. For simplicity and to emphasize the unifying character of this point of view, these examples are mostly for the traditional number theoretical semigroup N and the spaces R or C. 1. Introduction The “M¨obiusInversion Formula" in elementary number theory most often refers to the formula X X n (1.1) fb(n) = f(d) () f(n) = µ(d)fb ; d djn djn where f is an arithmetical function, that is, a function on N with values typically in Z, R or C; the sum ranges over the positive divisors d of a given 2000 Mathematics Subject Classification: Primary 11A25. Keywords: M¨obiusfunction, M¨obius transform, Dirichlet convolution, inversion formula, arithmetical semigroup, flow. 64 M. Benito, L. M. Navas and J. L. Varona n 2 N, and µ is of course the M¨obiusfunction, given by 8 µ(1) = 1; <> (1.2) µ(n) = 0 if n has a squared factor, > k : µ(p1p2 ··· pk) = (−1) when p1; p2; : : : ; pk are distint primes.
    [Show full text]